Dr. Z’s Math251 Handout #13.3 [Arc Length and Curvature]By Doron ZeilbergerProblem Type 13.3a: Find the length of the curve r(t) = x(t)i + y(t)j + z(t)k , t0≤ t ≤ t1.Example Problem 13.3a: Find the length of the curve r(t) = t2i + 2t j + ln t k , 1 ≤ t ≤ e.Steps Example1. Compute the derivativer0(t) = x0(t)i + y0(t)j + z0(t)k .1.r0(t) = (t2)0i + (2t)0j + (ln t)0k= 2t i + 2 j +1tk .2. Find the magnitude of r0(t),|r0(t)| =px0(t)2+ y0(t)2+ z0(t)2,and use algebra and/or trig to simplify asmuch as you can.2.|r0(t)| =r(2t)2+ 22+1t2=r4t4+ 4t2+ 1t2=r(2t2+ 1)2t2=(2t2+ 1)t= 2t+1t.3. Integrate the expression that you gotin step 2 from t0to t1.Zt1t0|r0(t)|dt3.Ze1|r0(t)|dt =Ze1[2t+1t] dt == t2+ln te1=e2+ln e−(12+ln 1) = e2+1−(1+0) = e2.Ans.: The arc length of that curve is e2.Problem Type 13.3b: Reparametrize the curve with respect to arc length measured from thepoint when t = t0in the direction of increasing t.r(t) = x(t) i + y(t) j + z(t) k .Example Problem 13.3b: Reparametrize the curve with respect to arc length measured from1the point when t = 0 in the direction of increasing t.r(t) = 5 sin t i + 3 j + 5 cos t k .Steps Example1. Compute r0(t), and then take its mag-nitude |r0(t)|.1.r0(t) = (5 sin t)0i + 30j + (5 cos t)0k .(5 cos t) i − (5 sin t) k .So|r0(t)| =p(5 cos t)2+ (−5 sin t)2=q25(cos2t + sin2t) = 5 .2. Integrate it from t0to t1. Get an ex-pression in terms of t1and call it s. Nowchange t1into t. Now solve for t in termsof s.2.s =Zt105 dt = 5t1.Changing the t1into t we gets = 5tand expressing t in terms of s, we gett = s/5 .3. Go back to the original r(t) and replacet by the expression in s that you found instep 2.3.r(t) = 5 sin t i + 3 j + 5 cos t k .becomes5 sin(s/5) i + 3 j + 5 cos(s/5) k .This is the Ans..2Problem Type 13.3c: Find the curvature forr(t) = x(t) i + y(t) j + z(t) k .Example Problem 13.3c: Find the curvature forr(t) = t i + 2t j + t2k .Steps Example1. Compute r0(t) and r00(t) . 1.r0(t) = i + 2 j + 2t k .r00(t) = 2 k .2. Compute the cross productr0(t) × r00(t).2. r0(t) × r00(t) equalsi j k1 2 2t0 0 2=i2 2t0 2− j1 2t0 2+ k1 20 0= 4 i − 2 j3. Find the magnitude of the vector thatyou found in step 2 (namely r0(t)×r00(t)).Also find the magnitude of r0(t), and fi-nally use the formula for the curvatureκ(t) =|r0(t) × r00(t)||r0(t)|33.|r0(t) × r00(t)| =p42+ 22+ 02=√20 .|r0(t)| = |i + 2 j + 2t k|=p12+ 22+ (2t)2=p5 + 4t2.Finally,κ(t) =√20(√5 + 4t2)3This is the